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Topic: Vectors summing to zero
Replies: 24   Last Post: Sep 9, 2010 8:48 AM

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 Ki Song Posts: 549 Registered: 9/19/09
Re: Vectors summing to zero
Posted: Sep 2, 2010 6:12 PM

On Sep 2, 7:09 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
wrote:
>
> As I go over it again I will stand by the 2 pi claim. If you have a
> falsification then please present it. I don't necessarily believe all
> of mainstream mathematics so I find it acceptable that you do not
> believe mine. Simply stated: the sum of the angles between three
> coplanar vectors will always be 2 pi, in any dimension. The converse
> holds: If the sum of the angles of three vectors is 2 pi, then the
> vectors are coplanar. This is not a proof, but is an observation. I'm
> open to falsification, but doubt that you will find one.
>
>  - Tim

I don't really understand what you mean by "sum of the angles between
three coplanar vectors will always be 2 pi." I'm assuming you meant
dimension greater than or equal to 2 when you said "in any dimension."
This statement is trivially false if you just meant "three vectors
that that lie in a 2 dimensional subspace."(Take any vector X. Then,
take 2X and 3X.) Even if you meant "three vectors that span a 2-
dimensional subspace," it is trivially false. (Take two linearly
independent vectors, X,Y. Choose X+Y as the third vector. Angle
between X&Y, X&(X+Y), and Y&(X+Y) will not add up to 2Pi.)

So obviously, you mean something else. Exactly what do YOU mean by
three co-planar vectors?

The converse of the statement is also trivially false. Look at a
tetrahedron constructed from 4 equilateral triangles. Imagine that
one of the vertices is the origin, and imagine that the three edges
are vectors in R^3. What are the angles between the vectors? What is
the sum of the angles of these vectors? Are these vectors "coplanar"?

I agree with Gerry that perhaps it would be wise for you to learn a
bit of conventional mathematics before rejecting it. If you know some
algebra, you'd know that your polysign numbers is just a example of a
quotient ring.